# Reference to a Classical Regularity Theorem

(Edited)

## I need a reference to the following result:

If $$u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$$ satisfies $$\begin{cases} {\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ \\ u = 0 & {\rm on} \ B_1' \end{cases}$$

where

$$F \in C^{1,\beta}(B_1^+\times\mathbb{R}\times\mathbb{R}^{n+1};\mathbb{R}^{n+1}), \quad F_0 \in C^{0,\beta}(B_1^+\times\mathbb{R}\times\mathbb{R}^{n+1};\mathbb{R})$$

for some $$0<\beta<1$$, and

$$\langle D_p F(x,u,p) \xi,\xi \rangle \ge \lambda(M) |\xi|^2$$

for some $$0 < \lambda(M) < + \infty$$, for every $$x \in \overline{B_1^+}$$, $$u \in \mathbb{R}$$, and $$|p| \le M$$,

then $$u \in C^{2,\alpha}(\overline{B_{1/2}^+})$$ for some $$0<\alpha<1$$.

## Notations:

$$B_1^+ = \{x = (x',x_{n+1}) \in \mathbb{R}^{n+1} : |x| < 1, \, \, x_{n+1} > 0\}$$ is the half-ball and $$B_1' = \{x = (x',0) \in \mathbb{R}^{n+1} : |x'| < 1\}$$ is the flat part of its boundary.
Also, we have $$n \ge 1$$.
$$H^2$$ denotes the Sobolev Space of functions with second order weak derivatives in $$L^2$$ and $${\rm Lip}$$ is the space of Lipschitz-continuous funcions, whilst $$C^{k,\alpha}$$ is the space of functions whose $$k$$-th order classical derivatives are Hölder-continuous of exponent $$\alpha$$.

The discussion from Section 13.1 in the book of Gilbarg and Trudinger shows that $$u \in C^{1,\,\alpha}\left(B_{3/4}^+\right)$$. From here one can apply Schauder estimates for linear equations. For example, one can pass the divergence on the left hand side and view $$u$$ as a solution to a non-divergence form linear equation with Hölder continuous coefficients (namely $$F^i_j(\nabla u)$$, in the case that $$F$$ depends only on $$\nabla u$$). For the relevant linear theory, see e.g. Section 5.5 from the book of Giaquinta and Martinazzi here.